Subtract $ \dfrac{5}{8} $ from $ \dfrac{-5}{3} $ to get $ \dfrac{ \color{purple}{ -55 } }{ 24 }$.

To subtract fractions they must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 8 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{-5}{3} - \frac{5}{8} & = \frac{ \left( -5 \right) \cdot \color{blue}{ 8 }}{ 3 \cdot \color{blue}{ 8 }} - \frac{ 5 \cdot \color{blue}{ 3 }}{ 8 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -40 } }{ 24 } - \frac{ \color{purple}{ 15 } }{ 24 }=\frac{ \color{purple}{ -55 } }{ 24 } \end{aligned} $$